Grind down concrete floor to get more ceiling height?

We plan on converting an attached garage into two bedrooms and a mixed-used area. It's about 700 sqft total. The issue is with ceiling height. From concrete floor to bottom of existing ceiling (5/8" plaster) is 7ft 5in. Code requires 7ft 4in for the finished space. That leaves me only 1" to seal and insulate the concrete and install a floor. I'd like a little more space, preferably so I can put another layer of ceiling drywall up for sound insulation.

So I can think of two options, both I'd have to contract out... just looking for some opinions:

1. Remove & Re-pour. I'd go about 8" deeper, add in thick insulation and a vapor barrier. Maybe even radiant. But that's going to set me back about $7000 not including the heat. I'm reasonably sure there are no structural issues and the concrete foundation footings are over 36" deep. House was built in the late 1940's.

2. Grind the floor down 1". How cost effective would this be, assuming I can find someone to do it? The floor is 1.5" thicker than the rest of the basement, so I'm pretty sure it could handle being 1" thinner. It's in very good shape now with no cracks but is dirty with oil and grease. Grinding would help get rid of the smell too. I've seen big buildings get it done with next to no dust, but I don't know if that's done on a residential scale.

Any thoughts? I hate to spend thousands of dollars to get an inch!! How would you guys approach this?

Grind down concrete floor to get more ceiling height?

Sorry for leaving out a critical detail.

The house is a ranch with a garage / walkout basement on the lowest level. We can't go up, since the existing main house is on top. As part of the basement refinish, we're converting the garage into bedrooms and extra living space.

If we remove the floor and dig... not sure I understand about conventional framing? Where would that be needed? The top of the footers are about 6" under the concrete. Not sure how wide they are inside, so we'd have to take that into account if we re-poured. Would it be permissible to have a small ledge or bump out near the floor if we poured lower than the footer top?